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Regular hexagon ABCDEF has a perimeter of 36. O is the center of the hexagon and of circle O. Circles A, B, C, D, E, and F have centers at A, B, C, D, E, and F, respectively. If each circle is tangent to the two circles adjacent to it and to circle O, what is the area of the shaded region (inside the hexagon but outside the circles)? (see attached file)

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hexagon:
if I don't know that the area of a regualr hexagon is (edge length)^2*(3/2)*sqrt(3)... (and why should I?)

...then I can figure it's made up of six equilateral triangles with sides of length 6.

by the pythagorean theorem, the height of each triangle is 3*sqrt(3)
(note each 60-60-60 triangle is 2 30-60-90 triangles and these have the old "one, two, radical three" ratio of sides we all loved in grade school, here the "two" is the length of the eqilateral triangle's side)

so the area is (1/2)*6*3*sqrt(3).

the area of the hexagon is then 6*(1/2)*6*3*sqrt(3) = 54*sqrt(3)

circles, all of radius 3, area 9pi:

the vertex angle of a regular hexagon is 120degrees.

so (120/360) or one third of each of six circles, plus the whole of the one circle in the middle of the hexagon, lies within the perimeter of the hexagon. three circles' area, all told.

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09 Aug 2006, 12:52

E it is!

Area of Hexagon - 3*(Area of Circle of radius 3)
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Uh uh. I know what you're thinking. "Is the answer A, B, C, D or E?" Well to tell you the truth in all this excitement I kinda lost track myself. But you've gotta ask yourself one question: "Do I feel lucky?" Well, do ya, punk?